There’s something rot ten in the state Educa tion Department. Year after year, New York officials have been claiming impressive gains in student achievement — claims we now know to be false. Based on these misleading claims, however, tens of millions of dollars in bonuses have been handed out to teachers and principals, and thousands of students have gotten false reports about their progress.

Last spring, the state triumphantly reported that 87 percent of New York fourth-graders were proficient in math. This was a striking improvement over 2006, when the state said that 78 percent had reached that goal.

The reported gains across the state were amazing. In Buffalo, the proportion of students in grades 3-8 who reached proficiency on the state math test leapt from 28.6 percent in 2006 to an incredible 63.3 percent in 2009; in Syracuse, from 30.1 percent to 58.2 percent — and in New York City, from 57 percent to 81.8 percent .

But on Wednesday, the federal government released the results of its independent testing program and produced very different results for New York.

The federal exams, the National Assessment of Educational Progress, are given to scientifically drawn samples of students in fourth and eighth grades in every state every other year.

Nationally, fourth-grade math scores on the NAEP were flat, while eighth-grade math scores were up significantly. In New York, however, there were no significant gains in either. The fabulous “gains” reported last spring, we now know, were based on dumbed-down tests and dubious scoring of the tests in Albany.

By federal standards, only 40 percent of our fourth graders are proficient, not 87 percent. And among eighth graders, only 34 percent are proficient in math, not the 80 percent claimed by the state.

The state Education Department also misled the public about progress in narrowing racial achievement gaps. The state claims that 78 percent of black students in fourth grade are proficient in math — but NAEP finds that only 19 percent are. In eighth grade, the state holds that 63 percent of black students are proficient; per NAEP, it’s just 13 percent.

Similarly, the state says that 82 percent of New York’s Hispanic fourth graders are proficient, while NAEP says it is 25 percent. In eighth grade, the state says 69 percent of Hispanic students are proficient, compared to 15 percent on NAEP.

The state says that 92 percent of white and 96 percent of Asian fourth-graders are proficient, but NAEP says that the real proficiency rate is 50 percent for whites and 67 percent for Asians. In eighth grade, the state reported 89 percent white proficiency and 92 percent among Asians, but just 44 percent and 63 percent were proficient by federal standards.

When the state announced those stunning “gains” last spring, critics wondered if the state had made the tests easier or manipulated the scoring. State officials dismissed the critics, insisting that the tests were even “harder” than past exams.

Heads should roll. Someone must be held accountable for this travesty against children, teachers, parents and taxpayers.

It’s time for the new brooms in Albany — Regents chief Merryl Tisch and Commissioner David Steiner — to fix the state’s flawed testing system.

Diane Ravitch is research professor of education at New York University. Her new book is “The Death and Life of the Great American School System.”